Functional inequalities for the decay of sub-Markov semigroups
Let \((E,{\mathcal F},\mu)\) be a measure space and \(P(t)\) a strongly continuous semigroup on \(L^2(\mu)\) with generator \(L\). The author investigates relationship between the following estimates: \[ \mu (P(t)f)^2) \leq \zeta(t) \Psi(f), \quad t > 0,\;f\in L^2(\mu), \] \[ \mu(f^2) \leq \alpha(r) \mu(fLf) +r\Phi(f), \quad r>0,\;f\in{\mathcal D}(L), \] where \(\zeta:(0,\infty)\to(0,\infty)\) is decreasing with \(\zeta(t)\to 0\) as \(t\to\infty\); \(\alpha:(0,\infty)\to(0,\infty)\) is decreasing; \(\Phi:L^2(\mu)\to[0,\infty]\), \(\Phi(cf)=c^2\Phi(f)\), \(c\in{\mathbb R}\); \(\Psi:L^2(\mu)\to[0,\infty]\), \(\Psi(cf)=c^2\Psi(f)\), \(c\in{\mathbb R}\). The results are illustrated by examples of diffusion and jump processes. Cheeger's inequality for the principal eigenvalue is generalized.
scientific article; zbMATH DE number 1901230